Stabilizability of linear impulsive systems

Abstract

This paper establishes the equivalence of three stabilizability-related properties for a class of linear impulsive systems. The first involves a gramian-based condition inspired by results for time-varying, discrete-time linear systems introduced decades ago. The second is the ability to achieve closed-loop exponential stability via state feedback. Finally, the third property is exponential stability of an 'unreachable' subsystem identified from a decomposition of the original system derived from an invariant subspace that characterizes the set of reachable states. A consequence of this analysis is that full state reachability of a linear impulsive system is not necessary for state feedback stabilization, a well-known fact for linear time-invariant systems. The main ideas of the paper are applied to the problem of synchronizing two Lorenz oscillators using underactuated impulsive control.

Stabilization on periodic impulse control systems

“…• About the stabilization for impulse control systems, we would like to mention [1,8,9,12,14,16] and the references therein.…”

“…Both results are related to some kind of reachability. The stabilization of the above system was also studied in [8].…”